1. Collatz Problem -- From MathWorld From Eric Weissten's World of Mathematics. Article with references and links.Category Science Math Number Theory Open Problems collatz problem......collatz problem, conjecture. Let be an integer. Then the collatz problemasks if iterating, (1). always returns to 1 for positive . http://mathworld.wolfram.com/CollatzProblem.html

Number Theory Sequences Collatz Problem A problem posed by L. Collatz in 1937, also called the mapping, conjecture . Let be an integer . Then the Collatz problem asks if iterating always returns to 1 for positive . The members of the sequence produced by the Collatz are sometimes known as hailstone numbers . Conway proved that the original Collatz problem has no nontrivial cycles of length . Lagarias (1985) showed that there are no nontrivial cycles with length . Conway (1972) also proved that Collatz-type problems can be formally undecidable The following table gives the sequences obtained for the first few starting values (Sloane's The numbers of steps the the algorithm to reach 1 for , 2, ... are 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, ... (Sloane's ). Of these, the numbers of tripling steps are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane's ), and the number of halving steps are 0, 1, 5, 2, 4, 6, 11, 3, 13, ... (Sloane's ). The smallest starting values of that yields a Collatz sequence containing n = 1, 2, ... are 1, 2, 3, 3, 3, 6, 7, 3, 9, 3, 7, 12, 7, 9, 15, 3, 7, 18, 19, ... (Sloane's

2. Collatz Problem Image By Andrew Shapira. The intensity of a point denotes the time taken to terminate.Category Science Math Number Theory Open Problems collatz problem......An Image From the collatz problem. By Andrew Shapira. February 15, 1998.(Minor We can do the same thing for the collatz problem. Given http://www.onezero.org/collatz.html

An Image From the Collatz Problem By Andrew Shapira February 15, 1998 (Minor revisions such as web link updates were made subsequently.) Introduction Consider the following rule that maps a given positive integer n to another: if n is even, the next integer is n/2 ; if n is odd, the next integer is . Starting at an arbitrary integer, we can repeatedly apply the rule to obtain a sequence of integers. For example: 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. It has been conjectured that all integers eventually yield a 1. The ``Collatz problem'', also known as the ``3x+1'' problem, is to determine whether the conjecture is true. The conjecture has been verified by computer up to . (See the table of contents at the sci.math FAQ and follow the link to ``Unsolved Problems.'') One day, Roddy Collins was showing me the Fractint package. Fractint is a package for generating images of fractals and fractal-like structures. Fractint has its own programming language, as well as a huge number of options for doing things like manipulating images and controlling parameters. The main operation in the programming language is to repeat a certain region of code until some termination condition is reached. The color or intensity at a given pixel corresponds to how many times the loop was iterated for the object that corresponds to the pixel. This reminded me of the Collatz problem, and I wondered whether we could use Fractint to draw a picture of the Collatz problem. I thought it would be neat to use the same kind of spiral pattern that has sometimes been used to graphically display prime numbers:

3. Ken Conrow Home Page The collatz problem (3x+1) I was introduced to the collatz problem back in 1990 by Dr. Ashok T. Amin here in the Computer Science Department at the University of Alabama in Huntsville. http://www-personal.ksu.edu/~kconrow

Ken Conrow Home Page Collatz 3n+1 Problem Structure I've finished putting up all the content I can think of concerning a structure I've developed about the Collatz 3n+1 problem. Mathematicians who refer to the problem as the problem were never brainwashed by FORTRAN (as I was) into the belief that n , not x , stands for an integer. I hope someone who can formalize mathematical proofs will see the potential here and take the appropriate set of ideas and sketch or complete a formal proof of the conjecture using them. You may communicate with me by e-mail at kconrow@ksu.edu . Constructive comments will be particularly welcome. If you want a quick trip through my work, look at a 18 slide slide show which contains a few pointers to illustrative material. Always use your browser's back button to return to the slide show if you look at some auxiliary material. One group of pages will be concerned with the main line of argument which I believe might lead to a proof of the Collatz problem. n +1 Problem Statement and References Synopsis/Abstract Index to Terms Used ... Proofs of Some Little Pieces of the Structure Another group of pages will just give miscellaneous auxilliary information which I find interesting for one reason or another but which seem not particularly relevant to a proof. Maple Provides the Clue in the Left Descent to 27 Showing Complete Paths and the Huge Scatter in the Instances of a Given Left Descent Assemblage Shortcut to Numeric Values of the Coefficients in the dn ... Density vs. Tree Path Conundrum

4. The 3x+1 Problem And Its Generalizations The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and http://www.cecm.sfu.ca/organics/papers/lagarias

The 3x+1 problem and its generalizations*** Jeff Lagarias Murray Hill, New Jersey Math activated text Other available formats Related links ... Author biography Abstract: (taken from the Introduction) The problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm , and Ulam's problem , concerns the behavior of the iterates of the function which takes odd integers n to and even integers n to n/2 . The Conjecture asserts that, starting from any positive integer n , repeated iteration of this function eventually produces the value The Conjecture is simple to state and apparently intractably hard to solve. It shares these properties with other iteration problems, for example that of aliquot sequences and with celebrated Diophantine equations such as Fermat's last theorem. Paul Erdos commented concerning the intractability of the problem: "Mathematics is not yet ready for such problems." Despite this doleful pronouncement, study of the problem has not been without reward. It has interesting connections with the Diophantine approximation of the binary logarithm of and the distribution mod 1 of the sequence , with questions of ergodic theory on the -adic integers, and with computability theory - a generalization of the

5. The Collatz Problem, Data And Models The collatz problem, related functions, data and models http://site.voila.fr/Collatz_Problem

6. On The 3x + 1 ProblemEric Roosendaal Presents This Eight Part Document On The 3x Jeffrey C. Lagarias writes this document on the 3x+1 problem also known as collatz problem and the Syracuse problem. Read an introduction and generalizations. Introduction. The 3x+1 problem. A heuristic argument. Do divergent trajectories exist? Connections of the problem to ergodic theory. http://personal.computrain.nl/eric/wondrous

7. Z-Number -- From MathWorld any Znumbers exist. The Z-numbers arise in the analysis of the Collatzproblem. collatz problem. References. Flatto, L. Z-Numbers http://mathworld.wolfram.com/Z-Number.html

Number Theory Special Numbers Miscellaneous Special Numbers Z -Number A Z -number is a real number such that for all k = 1, 2, ..., where frac is the fractional part of x . Mahler (1968) showed that there is at most one Z -number in each interval for integer n , and therefore concluded that it is unlikely that any Z -numbers exist. The Z -numbers arise in the analysis of the Collatz problem Collatz Problem References Flatto, L. " Z -Numbers and -Transformations." Symbolic Dynamics and its Applications, Contemporary Math. Guy, R. K. "Mahler's Z -Numbers." §E18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 220, 1994. Lagarias, J. C. "The Problem and its Generalizations." Amer. Math. Monthly http://www.cecm.sfu.ca/organics/papers/lagarias/ Mahler, K. "An Unsolved Problem on the Powers of 3/2." Austral. Math. Soc. Tijdman, R. "Note on Mahler's -Problem." Kongel. Norske Vidensk Selsk. Skr. Author: Eric W. Weisstein Wolfram Research, Inc.

8. AlDamen Chemistry student at Jerash University with interests in number theory and the collatz problem. http://www.angelfire.com/de2/abbas

9. Andrew Shapira's Home Page Andrew Shapira. Articles and Software. An Image from the collatz problem (the ``3x+1Problem''). An Introduction to Thanking after Ridiculously Fast Chess Games. http://www.onezero.org/

Andrew Shapira Articles and Software An Image from the Collatz Problem (the ``3x+1 Problem'') An Introduction to Thanking after Ridiculously Fast Chess Games The Worst Songs Ever Pictures Dogs Playing Poker (a PC game that 3 other guys and I wrote) My Healthy Food Tips andrew at onezero.org

10. International Conference On The Collatz Problem Katholische Universität Eichstätt, Germany; 56 August 1999. On-line proceedings and group photo.Category Science Math Number Theory Open Problems collatz problem......International Conference on the collatz problem and Related TopicsAugust 56, 1999 Katholische Universität Eichstätt, GERMANY. http://www.math.grin.edu/~chamberl/conf.html

International Conference on the Collatz Problem and Related Topics August 5-6, 1999 This conference is intended for anyone interested in the 3x+1 problem ( also known as the Syracuse algorithm, Collatz', Kakutani's, or Ulam's problem), and related mathematics. CONFERENCE SCHEDULE CONFERENCE PROCEEDINGS E-mail: xhillner@aol.com Phone: (08421) 982010 Fax : (08421) 982080 You may also want to see other places of accomodation ; click on the word "Tourist Info" and then "Hotels". REGISTRATION: US$60 or 54 Euro, payable at the conference. FINANCIAL SUPPORT: A limited amount of financial support may be available. The Willibaldsburg (castle) St. Peter's Dominican Church ORGANIZERS: Marc Chamberland Department of Mathematics Grinnell College Grinnell, Iowa 50112 U.S.A. Office: (515) 269-4207 Fax: (515) 269-4984 chamberl@math.grin.edu Germany Telefon: (08421) 93-1456 Telefax: (08421) 93-1789 guenther.wirsching@ku-eichstaett.de

11. Superset Homepage The limited halting problem finding machines that solve the halting problem for limited classes of inputs without reporting erroneous results. The collatz problem is a special case. Software, papers and graphics. http://www8.pair.com/mnajtiv/halt/halt.html

The Superset Algorithm By M.Najtiv, (Please send me mail if you wish to be added to the Superset email announcements and update list.) This site is dedicated to the "limited halting problem" : finding machines that solve the halting problem for limited classes of inputs without reporting erroneous results. The papers below should alone be sufficient to completely describe the overall ideas without the code. The quick-and-dirty code is available however. I hope that some people try to implement my algorithms without looking at my code for highly independent verification of results. I am studying the halting problem by attempting to solve the Collatz conjecture, or the 3n+1 problem (also called Ulam sequences or Hailstone numbers). Respondents have mentioned the Busy Beaver problem which I am not studying directly but have decided to compile some material on this site for interested researchers. the Superset Algorithm paper (Feb 97) - 41k, about 13 printed pages, posted to sci.math and comp.theory. Requires basic CS-undergraduate-level knowledge of automata theory. Describes a very promising approach to the limited halting problem based on Generalized Sequential Machines . Results suggest it might be at least sufficient to solve the Collatz conjecture (the "3n+1 Problem"). Describes how any Turing Machine can be reduced to a GSM and its halting properties potentially analyzed via elementary function and set membership principles . However, see caveat below on "gsm_superset" program.

12. Proceedings Of 3x+1 Conference Proceedings of the International Conference on the collatz problemand Related Topics. Some of the participants of the conference http://www.math.grin.edu/~chamberl/conference/proceedings.html

Proceedings of the International Conference on the Collatz Problem and Related Topics Some of the participants of the conference offer either a paper or abstract relating to their talk at the conference. The (3x+1)/2 Problem and its generalisation: a stochastic approach (postscript) M.R. Feix and J.L. Rouet Some Properties of the 3n+1 Function Using Number Representation (postscript) R. Banerji The 3x+1 Problem and Directed Graphs (postscript) P. Andaloro A Dynamical Systems Approach to the 3x+1 Problem (postscript) M. Chamberland The 3n+1-Problem and Holomorphic Dynamics (postscript) D. Schleicher A Category of Topological Spaces Encoding Acyclic Set-Theoretic Dynamics (postscript) (pdf) K. Monks Some Results on the Collatz Problem (postscript) M. Kudlek Left to right: K. M. Monks, K. G. Monks, P. Andaloro, G. Wirsching, M. Kudlek, R. Banerji, J. Lagarias, D. Schleicher, M. Chamberland, J. L. Rouet, E. Roosendaal, U. Fitze, M. R. Feix, E. G. Belaga

13. Collatz Problem exist a. collatz problem. Take any natural number m 0. n=m; repeatif (n is odd) then n=3*n+1; else n=n/2; until (n1). The conjecture http://db.uwaterloo.ca/~alopez-o/math-faq/node61.html

Next: Goldbach's conjecture Up: Unsolved Problems Previous: Does there exist a Collatz Problem Take any natural number m n m repeat n is odd) then n n +1; else n n until ( n The conjecture has been verified for all numbers up to References Unsolved Problems in Number Theory. Richard K Guy. Springer, Problem E16. Elementary Number Theory. Underwood Dudley. 2nd ed. G.T. Leavens and M. Vermeulen 3x+1 search programs Comput. Math. Appl. vol. 24 n. 11 (1992), 79-99. Alex Lopez-Ortiz Mon Feb 23 16:26:48 EST 1998

14. Famous Problems In Mathematics that is perfect and odd? collatz problem; Goldbach's conjecture; Twinprimes conjecture. Alex LopezOrtiz Mon Feb 23 162648 EST 1998. http://db.uwaterloo.ca/~alopez-o/math-faq/node55.html

Next: The Four Colour Theorem Up: Frequently Asked Questions in Previous: Names of Large Numbers Famous Problems in Mathematics The Four Colour Theorem The Trisection of an Angle Which are the 23 Hilbert Problems? Unsolved Problems ... Twin primes conjecture Alex Lopez-Ortiz Mon Feb 23 16:26:48 EST 1998

15. The Collatz Problem (3x+1) The collatz problem (3x+1). I was introduced to the collatz problemback in 1990 by Dr. Ashok T. Amin here in the Computer Science http://home.hiwaay.net/~criswell/math/collatz.html

The Collatz Problem (3x+1) I was introduced to the Collatz problem back in 1990 by Dr. Ashok T. Amin here in the Computer Science Department at the University of Alabama in Huntsville. Dr. Niall Graham, also here in the department, has recently revived my interest in it. The problem deals with sequences of integers generated as follows: Start with a positive integer x > 0. Repeat the following steps: If the last integer in the sequence is 1, stop. The sequence is complete. If the last integer in the sequence is even, divide it by two to get the next integer in the sequence. If the last integer in the sequence is odd, multiply it by three and add one to get the next integer in the sequence. The problem is very simple to state, and the actions are very simple to perform, but the question is, given any starting integer x > 0, will the sequence generated end with the integer 1 in a finite number of steps? Here are the sequences generated for the first few integers: Here is, perhaps, a neater way of showing it: (under construction) As you can see, they all end up at 1. It is interesting to turn this problem around and look at it in reverse, starting with 1 and going in reverse to produce sequences. The reverse of the procedure above is the following:

16. The 3x + 1 Problem And Its Generalizations A survey article by Jeff Lagarias.Category Science Math Number Theory Open Problems collatz problem...... http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html

Contents Next: Introduction The Problem and its Generalizations Jeffrey C. Lagarias Murray Hill, NJ 07974 (January 16, 1996) A list of math activations A linked list of keywords ... Contents Next: Introduction

17. Editing And Debugging M-Files (Development Environment) Debugging ExampleThe collatz problem. The collatz problem is to prove thatthe Collatz function will resolve to 1 for all positive integers. http://www.mathworks.com/access/helpdesk/help/techdoc/matlab_env/edit_d21.shtml

Development Environment Debugging ExampleThe Collatz Problem The example debugging session requires you to create two M-files, collatz.m and collatzplot.m , that produce data for the Collatz problem. For any given positive integer, n , the Collatz function produces a sequence of numbers that always resolves to 1. If n is even, divide it by 2 to get the next integer in the sequence. If n is odd, multiply it by 3 and add 1 to get the next integer in the sequence. Repeat the steps until the next integer is 1. The number of integers in the sequence varies, depending on the starting value, n The Collatz problem is to prove that the Collatz function will resolve to 1 for all positive integers. The M-files for this example are useful for studying the problem. The file collatz.m generates the sequence of integers for any given n . The file collatzplot.m calculates the number of integers in the sequence for any given integer and plots the results. The plot shows patterns that can be further studied. Following are the results when n is 1, 2, or 3.

18. Editing And Debugging M-Files (Development Environment) functions using an example. Closing MFiles, Debugging ExampleThecollatz problem, The MathWorks, Inc. Trademarks Privacy Policy. http://www.mathworks.com/access/helpdesk/help/techdoc/matlab_env/edit_d20.shtml

Development Environment Debugging M-Files This section introduces general techniques for finding errors, and then illustrates MATLAB debugger features found in the Editor/Debugger and equivalent debugging functions using a simple example. It includes these topics: Types of Errors Finding Errors Debugging ExampleThe Collatz Problem Trial Run for Example ... Using Debugging Features In addition to the Debugger and debugging functions, the Profiler included with MATLAB can be a useful tool to help you improve performance and detect problems in your M-files. For details, see Measuring Performance in the Programming and Data Types section of the MATLAB documentation. Types of Error s Debugging is the process by which you isolate and fix problems with your code. Debugging helps to correct two kinds of errors: Syntax errorsFor example, misspelling a function name or omitting a parenthesis. Syntax Highlighting helps you identify these problems, as does the process of setting breakpoints. When you run an M-file with a syntax error, MATLAB will most likely detect it and display an error message in the Command Window describing the error and showing its line number in the M-file. Click the underlined portion of the error message, or position the cursor within the message and press Ctrl+Enter . The offending M-file opens in the Editor, scrolled to the line containing the error. Use the

19. The Complexity Of The Collatz Problem The Complexity of the collatz problem The collatz problem is a verysimple, wellknown and unresolved problem of number theory. http://www.geocities.com/CapeCanaveral/Lab/4430/collatz.html

The Complexity of the Collatz problem The Collatz problem is a very simple, well-known and unresolved problem of number theory. It can be expressed like this: 1. Take any integer number. 2. Divide it by 2. If the division is exact, repeat step 2. 3. If it isn't, multiply it by 3, add 1 and go to step 2. For example, if you start with 7, you'll get: The question is: does this journey always end with 1? Computers have calculated this for numbers up to millions, and they've always ended at 1. But it has never been proven it has to be so for every number. Many mathematicians have attacked the problem with no result. Legend says scientists in Los Alamos spent a good deal of their time with it, instead of working in the atomic bomb! It was even rumored it was a Russian sabotage. I haven't solved it, I've got no idea about how to solve it, but I have a good insight of how complex it can be. Let's consider a generalized version of the problem: 1. Take any Gauss integer 2. Divide it by another called a . If the division is exact, repeat step 2. 3. If it isn't, multiply it by

20. A Heuristic Argument For The Collatz Problem A Heuristic Argument for the collatz problem. By. Joseph L. Pe. iDEN SystemEngineering Tools and Statistics. Motorola. 21440 West Lake Cook Road. http://www.geocities.com/SoHo/Exhibit/8033/collatz/collatz.htm

A Heuristic Argument for the Collatz Problem Joseph L. Pe iDEN System Engineering Tools and Statistics Motorola 1501 W. Shure Drive Arlington Heights, IL 60004 ajp070@motorola.com ABSTRACT. This paper offers, in support of the Collatz conjecture, a simple heuristic argument that appears to be easier than similar arguments that have been presented elsewhere. In the process, it also argues that, for sufficiently long initial segments of a Collatz sequence, the ratio of even to odd terms is bounded from below by the golden ratio. 1. Introduction Consider the Collatz function defined by f(n) = n/2 if n is even, and = 3n+1 if n is odd. It has been verified up to around n = 10 that the sequence n, f(n), f(f(n)), .... that is, the trajectory at n , eventually reaches 1. For example, the corresponding sequence for n = 12 is: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. The Collatz conjecture or problem , one of the most famous unsolved problems in number theory, states that the trajectory at n reaches 1 for all n. It is the aim of this paper to present a heuristic argument for the validity of the Collatz conjecture. In particular, this argument seems to be simpler than the one given in [L]. The finite sequence f(n), f(f(n)), ...., 1 (if it exists), where the last term is the earliest "1", is called the

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